Isotropic Beams Research Articles

A total Lagrangian elasticity formulation for the nonlinear free vibration of anisotropic beams

The large amplitude free vibration response of isotropic and anisotropic beams is considered using a total Lagrangian description of the motion along with Ritz-based approximations to the dynamic statement of virtual work. Results are presented for both axial and bending modes and are compared with existing results in the literature using various formulations along with a finite element solution of the Euler–Bernoulli beam with Föppl–von Karman nonlinearity. Results from the present formulation include all nonlinear strain terms and give frequency ratios that are in excellent agreement with both Euler–Bernoulli and higher order beam theories for slender beams under small amplitudes but grow noticeably higher than those of other beam models as the beam becomes thick and the amplitude of vibration increases.

Comparison of classical and refined beam models applied on isotropic and FG thin-walled beams in nonlinear buckling response

In this work, nonlinear buckling analysis of thin-walled beams by two different models is investigated: first one, developed by MUL2 group, based on Carrera Unified Formulation, and THINWALL v.16 beam model based on classical Euler-Bernoulli-Vlasov beam theory. In former, the refined beam theories are obtained on the basis of Taylor and Lagrange-type expansions. Latter is performed in framework of updated Lagrangian formulation adopting nonlinear displacement field of thin-walled cross-section. Isotropic and FG beams are referred to. It is observed that numerical results obtained by these methods match very well.

Quasi-3D dynamic analysis of rotating FGM beams using a modified Fourier spectral approach

Rotating beams and blades made of functionally graded materials (FGMs) have broad application prospects in engineering. Nonclassical features, such as anisotropy material, Coriolis effect, and both the centrifugal softening and stiffening effects, make it more challenging for one-dimensional theories to accurately model the rotating FGM beams. This work presents a quasi-three-dimensional solution for the dynamic behaviors of the rotating FGM beams. The FGM beam, attaching to a rigid hub, is assumed to have a metallic core covered with two ceramic faces. The constraint between the hub and FGM beam is simulated by the penalty method, which transforms the boundary potential energy into a quantifiable form and simplifies the selection of the trial functions. The model is established by employing the Carrera unified formulation (CUF) and modified Fourier spectral approach (MFSA) in which the displacement variables of the FGM beam are constructed by the modified Fourier series expansion. The variational method is applied for the derivation of the governing equations of the rotating FGM beams, considering the centrifugal and Coriolis effects. Several examples including the isotropic and FGM beams under different rotating conditions are performed to check the effectiveness and precision of the developed formulation. Parameter studies are then implemented to investigate the influences of the rotation velocity, material parameter, presetting angle and hub-radius ratio on the three-dimensional vibration characteristics of the rotating FGM beams.

Effect of the Multi Vibration Absorbers on the Nonlinear FG Beam Under Periodic Load with Various Boundary Conditions

A semi-analytical method is used to study the effects of the multi vibration absorbers on the nonlinear functionally graded (FG) Euler-Bernoulli beam subjected to periodic load. The material properties of the beam are assumed to be continuously graded in the thickness direction. The governing equations of functionally graded beam are obtained based on the Hamilton's principle and these equations are solved by using the Rayleigh-Ritz method. To validate the results, comparisons are made with the available solutions for the natural frequencies of isotropic beam. The effects of the multi vibration absorbers and material parameters on the vibration response of functionally graded beamare investigated. For case study the effect of two symmetrical vibration absorbers is considered, these absorbers are applicable in some of the mechanic systems. In those systems, two absorbers are used close to the beginning and end of the structures instead of using them in the middle of these structures. By considering the industrial applications, it is shown that using the two symmetrical vibration absorbers with lower mass is close to the end of functionally graded beam is better than the middle of one. Also, the effect of different numbers of the vibration absorbers on the nonlinear functionally graded beam with simply supported boundary condition is considered. The results shown that increasing the number of vibration absorbers leads to decreasing the maximum deflection.

Geometrically non-linear forced vibrations of fully clamped functionally graded beams with multi-cracksresting on intermediate simple supports

The objective of this work is to investigate the non-linear forced dynamic response at large vibration amplitudes of fully clamped functionally graded beams containing a multiple open edge cracks and resting on intermediate simple supports. The theoretical model is based on the Euler-Bernoulli beam theory and the crack rotational spring model. The functionally graded beam properties are supposed to vary continuously through the beam thickness. A homogenization procedure based on the neutral surface approach taking into account the presence of the crack is developed to reduce the problem examined to that of an equivalent isotropic homogeneous multi-cracked beam. In the non-linear analysis, harmonic motion is assumed and the discretized expressions for the beam total strain and kinetic energies are derived by applying Hamilton’s Principle, so as to reduce the problem to a non-linear algebraic system solved using an approximate explicit method previously applied to various non-linear structural vibration problems. Considering the forced vibration case, the non-linear frequency response functions are obtained numerically in the neighbourhood of the predominant non-linear mode shape using a single mode approach. The effects of crack, the beam material gradient and the applied harmonic force are presented and discussed.

Static Analysis of Orthotropic Euler-Bernoulli and Timoshenko Beams With Respect to Various Parameters

In this study, deflections of orthotropic beams along the beam length are calculated by using static analysis according to Euler-Bernoulli and Timoshenko beam theories. Since the mechanical properties of the materials change as the orientation angle of fibers changes, the formulation is carried out using the equivalent Young’s modulus and the equivalent shear modulus. Orthotropic beams are modeled as isotropic beams by using equivalent moduli. Governing equations are derived. Two numerical examples with different orthotropic materials are given for different boundary and loading conditions. The effect of changing the orientation angle of the fibers on the deflection values is also considered. Orientation angle, material properties, length to depth ratio has been considered as parameters in the static analysis of orthotropic beams. Results are also compared with steel which is an isotropic material and presented in the form of tables and graphs which may be useful.

HALE wing experiments and computational models to predict nonlinear flutter and dynamic response

AbstractExperimental and numerical investigations into the linear and nonlinear aeroelastic behaviour of very flexible High Altitude Long Endurance (HALE) wings are conducted to assess the effect of geometrical nonlinearities on wings displaying moderate-to-large displacement. The study shows that the dynamic behaviour of wings under large deflection, and specifically the edgewise and torsion natural frequencies and modal characteristics, are largely affected by the presence of geometrical nonlinearities. A modular wing structure has been manufactured by rapid prototyping and it has been tested to characterise its dynamic and aeroelastic behaviour. At first, several simple isotropic cantilever beams with selected crosssections are numerically investigated to extract their modal characteristics. Experiments are subsequently conducted to validate the geometrically nonlinear dynamics behaviour due to high tip displacement and to understand the influence of the beam cross-section geometry. The structural dynamics and aeroelastic analysis of a very flexible modular selected wing is then investigated. Clean-wing wind-tunnel tests are carried out to assess flutter and dynamic response. The wind-tunnel model display interesting aeroelastic features including the substantial influence of the wing large deformation on its natural frequencies and modal characteristics.

A three-dimensional periodic beam for vibroacoustic isolation purposes

Abstract This paper presents results of investigations on a three-dimensional (3-D) isotropic periodic beam. The beam can represent a vibroacoustic isolator of optimised dynamic characteristics in the case of its longitudinal, flexural and torsional behaviour. The optimisation process concerned both the widths as well as the positions of particular frequency band gaps that are present in the frequency spectrum of the beam. Since the dynamic behaviour of the beam is directly related to its geometry, through an optimisation process of the beam geometry, desired dynamic characteristics of the beam were successfully obtained. For the purpose of the optimisation process a new numerical model of the beam, based on the spectral finite element method in the time domain (TD-SFEM), was developed by the authors. This model enabled the authors to investigate the beam behaviour not only in a wide frequency spectrum, but also ensured a high accuracy of the model predictions. The accuracy of this modelling approach was checked against well-known analytical formulas. However, in the case of the optimised geometry of the beam for the verification of the correctness of the modelling approach a commercial finite element method (FEM) package was used. Finally, based on the results of numerical predictions and optimised geometry of the beam a sample for experimental verification was prepared. Experimental measurements were carried out by the authors by the application of one-dimensional (1-D) laser Doppler scanning vibrometry (LDSV). The results of experimental measurements obtained by the authors confirmed the correctness of the numerical predictions, showing a high degree of correspondence.

Flexural Response of Cantilever Beam Subjected to Varying Load by HSDT

Later the revolution of different theories in the shear deformation theories, a Higherorder Shear Deformation Theory (HSDT) captivating into account transverse shear deformation effect has been grown; this will be worn for the static flexure study of thick isotropic beams. In these theories a displacement field is prolonged up to the second power of thickness coordinate of beams to have the parabolic difference of transverse shear stresses. The theory assume a parabolic deviation of transverse shear stress crosswise the thickness of the beams. A transverse shear stress will be obtain directly from the make use of constitutive relations, pleasing the shear stress-free boundary conditions at top and bottom of the beam. Governing differential equations and boundary conditions of theory will be obtain by the principle of virtual work. Common clarifications of thick isotropic simply cantilever beam subjected to various loading conditions will be analyzed for transverse displacement, axial displacement, transverse shear stress and axial stress. The outcomes of the contemporary theory will be compare with those of supplementary refined shear deformation theories of beam to authenticate the accuracy of the theory.

Identification of Non-stationary Load Upon Timoshenko Beam

This paper investigates an inverse non-stationary problem of the restoration of the spatial law of a homogeneous isotropic Timoshenko beam of finite length. Hinge support conditions are used as boundary conditions. Initial conditions are assumed to be zero. It is assumed that of the beam’s ends is fitted with sensors which in the course of corresponding experiment register the amount of deflection of the beam at the sensor points. The method of the solution of a direct problem is based on the principle of superposition where the deflection of the beam is associated with the space load the beam is exposed to, by means of an integral operator by the spatial coordinate and time. The kernel of such operator is so called influence function. This function is a fundamental solution of a system of differential equations of motion of the study beam. The construction of such solution represents a separate problem. The influence function is found by means of Laplace time transformation and expansion into Fourier series in a system of the problem’s eigenfunctions. The solution of the inverse problem at the first stage reduces to a system of algebraic equations for vector operator whose components are time convolutions of the coefficients of expansion series for an influence function with the desired coefficients of expansion of the load in a Fourier series. At the same time, the components of the vector of the rights parts are time dependencies registered by the sensors. The resulting system is ill-conditioned [1]. The second stage serves to resolve independent Volterra integral equations of the first kind for the desired coefficients of Fourier serials for the load.

NURBS-Based Isogeometric Analysis Method Application to Mixed-Mode Computational Fracture Mechanics

In this paper, unified shear deformation theory is used to analyze simply supported thick isotropic beams for the transverse displacement, axial bending stress, transverse shear stress and natural frequencies. This theory enables the selection of different in-plane displacement components to represent shear deformation effect. The numbers of unknowns are same as that of first order shear deformation theory. The governing differential equations and boundary conditions are obtained by using the principle of virtual work. The results of displacement, stresses, natural bending and thickness shear mode frequencies for simply supported thick isotropic beams are presented and discussed critically with those of exact solution and other higher order theories. The study shows that, while the transverse displacement and the axial stress are best predicted by the models 1 through 5 whereas models 1 and 2 are overpredicts the transverse shear stress. The model 4 predicts the exact dynamic shear correction factor (pi^2/12 = 0.822) whereas model 1 overpredicts the same.

Construction of new enriched beam models accounting for cross-section deformation and pinching

The consideration of nonclassical beam theories beyond Timoshenko beam model is necessary in applications involving complex yarns subjected to transverse compaction, for which the section dilatation and in-plane shear require a detailed kinematic and static modeling. In the present work, new enriched beam elements with additional degrees of freedom are derived from the mechanics of generalized continua, allowing a representation of the in plane-shear and dilatation of sections. As a novel aspect, enriched beam model called macrodilatation, macroshear, and macrostrain have been constructed on the basis of generalized beam theories but with a more condensed kinematics, assuming uniform hyperstress tensor within the beam. The choice of the adequate extended continuum for a given beam depends on its macroscopic deformation mechanisms. As a result, the developed beam models include modified material parameters (the tensile, shear and bulk moduli for an isotropic beam) involving additional microstructural parameters. These macrobeam models are expected to better describe the in plane-shear and dilatation of the beam sections along its mean line in comparison to classical beam theories. Numerical illustrative examples show that the microdilatation beam model has the ability to capture the local deformation field around holes, in contrast to Timoshenko beam model. The parameters of the microdilatation beam model provide a very good estimate of the overall porosity.

Clamped-free micro-scale transversely isotropic thermoelastic beam resonators

The design of smart sensors/actuators are significantly affected due to clamped-free boundary condition. An analytical expressions for the thermoelastic damping (TED) and frequency shift of homogenous, transversely isotropic thermoelastic micro-scale clamped-free beam with linearly varying thickness, based on Euler–Bernoulli theory, have been derived. The effect of length and thickness on the quality factor of TED has been explored in detail with the help of MATLAB programming software and computer simulated results illustrated for Silicon Nitride micro-beam resonators with linearly varying thickness.

Damage propagation in 2d beam lattices: 2. Design of an isotropic fault-tolerant lattice

The paper demonstrates a rational design of an isotropic heterogeneous beam lattice that is fault-tolerant and energy-absorbing. Combining triangular and hexagonal structures, we calculate elastic moduli of obtained hybrid heterogeneous structures; simulate the development of flaws in that composite lattice subjected to a uniform uniaxial deformation; investigate its damage evolution; measure various characteristics of damage that estimate fault tolerance; discuss the trade-off between stiffness and fault tolerance. A design is found that develops a cloud of small evenly spread flaws instead of a crack.

Buckling of laminated composite and sandwich beams due to axially varying in-plane loads

This paper is dedicated to study the elastic buckling behavior of isotropic, laminated composite and sandwich beams subjected to various axially varying in-plane loads and boundary conditions (BCs). The formulation of the problem is derived by using the Ritz method with the displacement field based on a shear and normal deformable beam theory (SNDBT). Polynomial functions are employed to present the displacement field. The convergence studies are performed and then obtained results are compared with those of reported works. Results from extensive analysis are presented for different BCs, aspect ratios, orthotropy ratios, fiber angles and loading conditions. It is observed that the type of the axially variable in-plane load significantly affects the critical buckling loads and mode shapes of the beams depending on the BCs. The normal deformation effect depends on not only the aspect ratio but also BCs and the fiber orientation angles.

Modelling of the interaction between nonlinear solitary waves and composite beams

We examine numerically the interaction between solitary waves in lightly pre-compressed granular chains and laminated composite beams. A discrete element (DE) model is developed to simulate, with low computational effort, the propagation of solitary waves in a linear array of spherical particles as well as their interaction with a transversely isotropic composite beam. The derived equations allow identifying the governing dimensionless parameters and their effects on the delays and amplitudes of the reflected solitary waves are studied. It is found that the formation of reflected solitary waves is highly sensitive to the beam's geometry as well as to the elastic properties and layup of the laminate. For a beam of given dimensions, the delays and amplitudes of the reflected solitary waves are strongly affected by the flexural stiffness of the composite beam. The DE predictions are compared to those obtained from a more detailed hybrid discrete/finite element model and good agreement is reported for a wide range input parameters. The results of this study support the idea of developing a low-cost solitary wave-based diagnostic scheme for non-destructive evaluation of basic elastic properties of engineering composites.

Stochastic time domain spectral element analysis of beam structures

In this work, a stochastic time domain spectral element method (STSEM) is proposed for stochastic modeling and uncertainty quantification of engineering structures. To perform the analysis, both an isotropic Timoshenko beam (TB) and a sandwich beam are considered. The sandwich beam is modeled considering higher-order sandwich panel theory which is capable of addressing the core flexibility. The material properties are considered as 1D non-Gaussian random fields, and optimal linear estimation (OLE) is used for the discretization of the random fields. The OLE-based discretization of a random field allows simulating the random fields numerically, resulting in realizations of the stiffness, mass matrix, and dynamic stiffness matrix. In the current work, the computationally efficient time domain spectral element method (TSEM) is used to develop the STSEM formulation. The STSEM reduces the CPU time significantly as the number of degrees of freedom (dof) is much smaller than in FEM. TSEM also provides a consistent diagonal mass matrix which reduces the computation cost in case of dynamic problems. The deflection statistics of the beam for static, free vibration and dynamic cases are investigated for both TB and sandwich beam. The computational efficiency of the proposed method and the effect of material variability on the response statistics are also discussed. Moreover, the effect of different correlation lengths on the response statistics is studied.

Steeper dose gradients resulting from reduced source to target distance-a planning system independent study.

PurposeTo quantify the contribution of penumbra in the improvement of healthy tissue sparing at reduced source‐to‐axis distance (SAD) for simple spherical target and different prescription isodoses (PI).MethodA TPS‐independent method was used to estimate three‐dimensional (3D) dose distribution for stereotactic treatment of spherical targets of 0.5 cm radius based on single beam two‐dimensional (2D) film dosimetry measurements. 1 cm target constitutes the worst case for the conformation with standard Multi‐Leaf Collimator (MLC) with 0.5 cm leaf width. The measured 2D transverse dose cross‐sections and the profiles in leaf and jaw directions were used to calculate radial dose distribution from isotropic beam arrangement, for both quadratic and circular beam openings, respectively. The results were compared for standard (100 cm) and reduced SAD 70 and 55 cm for different PI.ResultsFor practical reduction of SAD using quadratic openings, the improvement of healthy tissue sparing (HTS) at distances up to 3 times the PTV radius was at least 6%–12%; gradient indices (GI) were reduced by 3–39% for PI between 40% and 90%. Except for PI of 80% and 90%, quadratic apertures at SAD 70 cm improved the HTS by up to 20% compared to circular openings at 100 cm or were at least equivalent; GI were 3%–33% lower for reduced SAD in the PI range 40%–70%. For PI = 80% and 90% the results depend on the circular collimator model.ConclusionStereotactic treatments of spherical targets delivered at reduced SAD of 70 or 55 cm using MLC spare healthy tissue around the target at least as good as treatments at SAD 100 cm using circular collimators. The steeper beam penumbra at reduced SAD seems to be as important as perfect target conformity. The authors argue therefore that the beam penumbra width should be addressed in the stereotactic studies.

Transversely isotropic poroviscoelastic bending beam solutions for low-permeability porous medium

This paper applies the recently completed theory of anisotropic poroviscoelasticity to simulate the three-point beam bending test on fluid saturated porous media, and to estimate materials’ low permeability and poroviscoelastic properties. The work completes the analytical solutions for this type of beam-bending tests, accounting for constant deflection or constant load together with three loading beam geometries to consider the anisotropic orientations, subjected to three sets of boundary conditions consistent with common laboratory set-ups. The analysis of the numerical example and the experimental data shows that the presented fully-coupled analytical solutions successfully capture the time characteristics of the force decay caused by pore fluid diffusion and the material relaxation. Material anisotropy also plays an essential role in the magnitude of the force decay. The permeability and the poroviscoelastic characteristics can be estimated by matching the analytical solutions of force decay or deflection increment with the experimental data.

Dynamics of a functionally graded Timoshenko beam considering new spectrums

Dynamics of a functionally graded (FG) beam were studied using Timoshenko and Euler-Bernoulli (EB) beam theories. Wave propagation of infinite beams and vibration analysis of simply supported FG beams were investigated. Variation of the material properties were assumed in the power law form. It was obtained that unlike isotropic beams, axial and transverse waves are coupled in FG beams due to unsymmetric material variation in the thickness direction of the beam. Two and three dispersion curves were obtained for EB and Timoshenko beam theory, respectively. All of these modes are axial-flexural coupled modes and coupling degree depends on material distribution with respect to mid-surface of the beam. The spectrums of different beams may be classified considering corresponding mode shapes. Wave propagation and vibration properties were discussed considering their mode shapes. It is seen that, unlike isotropic beams, pure shear mode is not possible for FG beams.